The speed of the wave along the Slinky depends on the mass of the Slinky itself and the tension caused by stretching it. Since both of these things have not changed, the wave speed remains constant.

The wave speed is equal to the wavelength multiplied by the frequency.

Since she is moving her hand faster, the frequency has increased. Since the velocity has not changed, an increase in the frequency would decrease the wavelength

The frequency at which standing waves are produced is known as the resonant frequencies. When two objects are brought near each other, and they both make standing waves at the same frequency, there is resonance in the system. For example, if you have two tuning forks of the same note, you can tap one and bring it close (but not touch) the other. Then if you silence the first tuning fork and listen, you will hear the second fork ringing as well because it vibrates at the same frequency.

This is an example of the doppler effect.  In this case the source is moving away from a stationary observer.  Therefore the corresponding equation is

When two waves are superimposed, the interference can be either constructive or destructive. In this case the interference is destructive, which means our resultant amplitude will be the difference of the two given amplitudes.

That means our new amplitude will be .

We can calculate the frequency of the beat using the equation

This is the beat frequency which means that the other string is off by   either higher or lower.

When we think about a violin or a guitar string that is being plucked, there are a variety of waves of different frequencies that will travel along the string and be reflected back at the ends.  Most interfere with each other and disappear.  The few that don’t will remain.  These are called the resonant frequencies. These frequencies create standing waves on the string.  If the frequencies do not align, they will interfere and die out.  However, if the frequency is the same standing waves occur and resonance occurs.  We can also see this in a tuning fork.  If we have two tuning forks of the same frequency and we strike the first and bring it near the second, it will begin to resonant as well as the air molecules strike the second fork causing it to vibrate.  If we do the same test with two different frequency forks, nothing will occur.

Beats occur due to the interference of two different frequencies that are occurring at the same time.  These beats are repetitive patterns of loud and soft interference that happen at regular intervals.  To calculate the beat frequency, simply take the difference of the two frequencies.  This means that if the higher frequency is increased, the beat frequency is increasing and is getting farther away from the lower frequency.  If the lower frequency is increased, the beat frequency is decreasing as it is getting closer to the higher frequency and therefore reducing the difference between the two.

The Doppler effect states that the frequency of sound increases as it approaches you and decreases as it goes away from you.

Since it is going away from you, the pitch of the sound will appear to get lower.

With the information provided we can determine the frequency of the beats.

This means that the string of the piano could either be this much higher or lower than the  string.

When two waves are superimposed, the interference can either be constructive or destructive. In this case, the interference is constructive and the waves are in phase, which means we add the amplitudes together.

Since each wave has an amplitude of , our new amplitude will be .